This is from a class question, not looking for someone to do homework for me, just seeking clarification.
Let the set X=(a,b,c)
How many distinct total orders can be defined on this set? Why?
I did search around a bit and from here, someone said that for a set of N elements the number of total orders is N!
My question is: is this correct, and why? My class notes define a "total order" as a partial order which is comparable, with a partial order being a relation that is Reflexive, Transititve and Antisymmetric. I get all that, but I'm confused as to how this relates to the question given.
Best Answer
Yes, $N!$ is correct.
Note that for your set {a,b,c} we get $$\{a<b<c, a<c<b, b<a<c, b<c<a, c<a<b, c<b<a\}$$
Thus these are just the arrangements with order which is permutations and the number of permutations on $N$ is $N!$